Derivatives of Trigonometric Functions We investigate lim

We investigatelimφ→0sin φ_φ (for simplicity assume 0 < φ < π/2) 1 1 φ A B C D sin φ E F a b α tan α = b_a b = a · tan α sin φ =|BD| < φ =⇒ sin φ φ <1

φ <|CE| + |EB| & |EB| < |EF| =⇒ φ <|CE| + |EF| = |CF| φ <|CF| = 1 · tan φ = sin φ

cos φ =⇒ cos φ <

sin φ φ <1

We use the Squeeze Theorem: lim

φ→0cos φ = 1 = limφ→01 =⇒ φlim→0

sin φ

We have the following identities for sin and cos: sin(x + y ) = sin x · cos y + cos x · sin y cos(x + y ) = cos x · cos y − sin x · sin y

We will prove that d dx sin(x ) = cos(x ) We have d dx sin(x ) = limh→0 sin(x + h) − sin(x ) h = lim h_→0

sin x cos h + cos x sin h − sin(x ) h

= lim

h→0

 sin x cos h − sin(x)

h + cos x sin h h  = lim h→0  sin xcos h − 1 h +cos x sin h h  =sin x · lim h→0 cos h − 1 h +cos x · hlim→0 sin h h | {z } 1

We will prove that d

dx sin(x ) = cos(x ) We have

d

dx sin(x ) = sin x · limh_→0

cos h − 1 h +cos x =sin x · lim h→0  cos h − 1 h · cos h + 1 cos h + 1  +cos x =sin x · lim h→0  (cos h)2₋₁ h(cos h + 1)  +cos x =sin x · lim h_→0  −(sin h)2 h(cos h + 1)  +cos x =sin x ·  lim h→0 sin h h · limh→0 −sin h cos h + 1  +cos x =sin x · 1 · 0 + cos x = cos x

d dx sin x = cos x d dx cos x = − sin x Differentiate f (x ) = x2sin x We have f0(x ) = x2 d dx sin x + sin x d dxx 2 _{product rule} =x2cos x + 2x sin x

d dx sin x = cos x d dx cos x = − sin x Differentiate tan x : d dx tan x = d dx  sin x cos x  = cos x · d dxsin x − sin x · d dxcos x (cos x )2

= cos x · cos x − sin x · (− sin x ) cos2_x = cos 2_{x + sin}2_x cos2_x = 1 cos2_x =sec 2_x

d

dx sin x = cos x

d

dx cos x = − sin x Differentiate thesecant sec x = _{cos x}1 :

d dx sec x = d dx  1 cos x  = cos x · d dx1 − 1 · d dxcos x (cos x )2 = sin x

d

dx sin x = cos x

d

dx cos x = − sin x Differentiate thecosecant csc x = _{sin x}1 :

d dx csc x = d dx  1 sin x  = sin x · d dx1 − 1 · d dxsin x (sin x )2 = −cos x sin2x = −csc x · cot x

Summary: derivatives of trigonometric Functions d dx sin x = cos x d dx cos x = − sin x d dx tan x = 1 cos2_x =sec 2_x d dx cot x = − 1 sin2x = −csc 2_x d

dx sec x = sec x · tan x

d

Differentiate f (x ) = sin(x2).

We have f = g ◦ h where g(x ) = sin x and h(x ) = x2 : g0(x ) = cos x

h0(x ) = 2x

f0(x ) = g0(h(x )) · h0(x ) = cos(x2)· 2x = 2x cos(x2)

Differentiate g(x ) = sin2x = (sin x )2.

We have f = g ◦ h where g(x ) = x2 and h(x ) = sin x : g0(x ) = 2x

h0(x ) = cos x

Differentiate f (x ) = esin x_.

We have f = g ◦ h where g(x ) = ex and h(x ) = sin x : g0(x ) = ex

h0(x ) = cos x

Differentiate f (x ) = sin(cos(tan x )). f0(x ) = cos( cos(tan x ) ) · d dx cos(tan x ) =cos(cos(tan x )) · (− sin(tan x )) · d dx tan x = −cos(cos(tan x )) · sin(tan x ) · 1 cos2_x